A projectile is fired from the origin with angle of elevation and initial speed v 0

Question:

A projectile is fired from the origin with angle of elevation α and initial speed v₀. Assuming that air resistance is negligible and that the only force acting on the projectile is gravity, t, that the position vector of the projectile is

We also showed that the maximum horizontal distance of the projectile is achieved when α = 45° and in this case the range is R = v²₀/g.

(a) At what angle should the projectile be fired to achieve maximum height and what is the maximum height?

(b) Fix the initial speed v0 and consider the parabola x² + 2Ry − R² − 0, whose graph is shown in the figure at the left. Show that the projectile can hit any target inside or on the boundary of the region bounded by the parabola and the x-axis, and it can’t hit any target outside this region.

(c) Suppose that the gun is elevated to an angle of inclination α in order to aim at a target that is suspended at a height h directly over a point D units downrange. The target is released at the instant the gun is fired. Show that the projectile always hits the target, regardless of the value v₀, provided the projectile does not hit the ground “before” D.